One clever pattern solves this entire exponent equation.

To solve this, you'll need to have a solid understanding of your exponential rules, quadratic equations, and how to manipulate them. Welcome back to my channel. In this particular equation, we are asked to find the value of m. Okay, first, looking at this, 2 to the power of m plus 2 to the power of 2m can be written as m raised to the power of 2, right? That is because a to the power of m times n is equal to a to the power of m raised to the power of n. It is also equal to a to the power of n, raised to the power of m. Okay, so both can be interchanged. So if we do that, that is why I can say this is 2 raised to the power of m, raised to the power of 2. Similarly, we can then also write this as 2 raised to the power of m raised to the power of 3. And everything is 84. Are you seeing something? But before we get into that, if this is your first time encountering my channel, Please hit the subscription button and turn on your notification bell so you never miss my videos, okay? I upload my videos as often as possible by the grace of God. So tune in at your convenience. There must be something new for you to watch. Give us a thumbs up if you love what we're doing. And please, in your caption, in your caption, look for this. On it. If my accent is too strong for you to understand. And let me know in the comments if you'd like me to auto-dub my videos. That is to translate it using AI into different languages. Just let me know in the comments and I'll give it a try again. Because I've done that before, and some of my subscribers complained that the AI was giving them incorrect information about what I was saying, okay? Alright, thank you. So back to what I was saying or what we were discussing. We can see that we have 2 to the power of m here, 2 to the power of m here, and 2 to the power of m here. Let's simplify our equation. So we can say let 2 raised to the power of m be equal to x. In that case, our equation becomes x plus x squared plus x cubed equals 84. Now let's divide this 84 into three numbers. And each number is going to be both a square and a cube of a particular number. So we have x plus x squared plus x cubed equals 64 plus 16 plus 4. How do I know? Because I know that 4 squared is 16 and 4 to the power of 3 is 64. Similar to what we have here. So now I have x plus x squared plus x cubed equals, this is 4 cubed, this is 4 squared, and this is 4. Alright, moving forward, let's bring this together and group them together. So we have x and minus 4, plus we have x squared and minus 4 squared, plus we have x to the power of 3 minus 4 to the power of 3. And everything is now equal to zero. So if we solve for that, remember that a squared minus b squared equals a minus b multiplied by a plus b. Similarly, a to the power of 3 minus b to the power of 3 is the same thing as a minus b multiplied by a squared plus ab plus b squared. So this is exactly how we are going to expand this now. So because of that, I'll need a lot of space, so I'll be cleaning this. Let me check my, it's still on. So let's remember that 2 raised to the power of m is equal to x. Now I will be cleaning everything. Alright, now take a look at what we have here. So we can write x minus 4 plus. Now, instead of this, I will write it in this form: x minus 4, closed bracket, x plus 4. Okay, plus, we open our bracket. Now, instead of writing this in that manner, I will expand it in this format. That will give me, I need to clean this. So for this, it will now be x minus 4 times the quantity x squared plus 4x plus 4, all squared. Okay, and everything will now be zero. Now let's solve: we have x minus 4 plus x minus 4, closed bracket x plus 4 plus x minus 4. I need to take that breath. x squared plus 4x plus 16 equals 0. Now we have x minus 4, x minus 4, x minus 4. Therefore, we can write it out and factorize it. So本赛季 itself is 1. This into this, this will remain. Is equal to 0. So we have x minus 4 into 1 plus 4, 5. 5 plus 16 is 21. x plus 4x, 5x plus x squared equals 0. Okay, now I think we are almost done. At this point, we can actually see that we have two equations here. We can form from here, right? Let me clean this side. So the two equations are x minus 4 equals 0. Or 21, or let me write this as a quadratic equation, supposed to be x squared plus 5x plus 21 equals 0. So for this first equation, our x is 4, right? Now, for this, if we solve it using the quadratic equation formula, which says x equals minus b plus or minus the square root of b squared minus 4ac. Everything divided by 2a. Let me focus on this place first. If we take the square root of b squared minus 4ac. We are going to have the square root of our b equal to 5. Five squared minus four times our a is 1. Our c is 21. So we are going to have the square root of 25 minus 84. That means we are going to get a number less than 0. So we are going to obtain complex values here. And I don't need that. I just need whole numbers. Therefore, we are going to skip this and go with this. Our x is equal to 4. So recall that 2 to the power of m is equal to x. So it means 2 to the power of m equals 4. Since our x is 4, right? So 2 to the power of m is equal to... Let's convert 4 to base 2, which is 2 squared. Now, since they share the same base, it simply means that their powers are equal. So it means m is also equal to 2. Now let's check. 2 to the power of 2 plus 2 to the power of 2 multiplied by 2. I am substituting m with 2. 2 raised to the power of 3 multiplied by 2. Is this 84? We are still checking. So this is 4. 2 to the power of 4 is 16. 2 to the power of 5 is 32. This is 3 times 2, which is 6. So 2 to the power of 6 is actually 64. So if we add 64 and 16, it will give us 80. 80 plus 4 is 84. That indicates our answer is correct. Thank you so much for watching. See you in my next video. Bye.